364 



THE POPULAR SCIENCE MONTHLY. 



ings of each other and following the explanations. There con- 

 tinually come up at these times incidents of this sort : Fig. 13 is 

 given as a solution of " Can you raise a perpendicular to a line, 

 and from the end of it ? " 



In his explanation the pupil points out the given line, the end 

 from which he is to erect the perpendicular, the point from which 

 as a center he sweeps each circle, why he may take that point, and 



why he sweeps the circle. Another pupil discovers, before the ex- 

 planation is finished, that the problem can be solved with one less 

 circle, and there is the keenest interest while he draws and ex- 

 plains his way (Fig. 14). 



The original and independent power acquired is shown in Figs. 



J 



15, 16, 17, and 18, solutions of the problem, " Can you make an oc- 

 tagon with one side given ? " 



Spencer's " Inventional Geometry " is one of the most original 



